Equal Leg Angles

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Equal Leg Angles (L) are L-shaped sections with equal length legs, versatile for bracing, trusses, and connections.

Calculations

Description

Symbol Name

Value

Unit

Comment

Equal Leg Angles (L)

Geometru Input

Side length

h_{t}

200.00

mm

Thickness

t_{t}

24.00

mm

Root radius

r_{r o o t}

18.00

mm

Toe radius

r_{t o e}

9.00

mm

Material Properties

Young's modulus

E_{t}

210000

\frac{N}{{mm}^{2}}

Poisson's ratio

v_{t}

0.30000

-

Density

ρ_{t}

7850.00

\frac{kg}{m^{3}}

Shear modulus

G_{t}

80769.2

\frac{N}{{mm}^{2}}

G_{t} = \frac{E_{t}}{(2 \cdot (1 + v_{t}))} = \frac{210000}{(2 \cdot (1 + 0.30000))}

Section Properties

A_{web}

4224.00

{mm}^{2}

A_{web} = (h_{t} - t_{t}) \cdot t_{t} = (200.00 - 24.00) \cdot 24.00

A_{flange}

4800.00

{mm}^{2}

A_{flange} = h_{t} \cdot t_{t} = 200.00 \cdot 24.00

A_{toe,r}

-17.38

{mm}^{2}

A_{toe,r} = - {r_{t o e}}^{2} \cdot (1 - \frac{π}{4}) = - {9.00}^{2} \cdot (1 - \frac{π}{4})

A_{root,r}

69.53

{mm}^{2}

A_{root,r} = {r_{r o o t}}^{2} \cdot (1 - \frac{π}{4}) = {18.00}^{2} \cdot (1 - \frac{π}{4})

Cross-sectional area

A_{t}

9058.77

{mm}^{2}

A_{t} = A_{flange} + A_{web} + A_{root,r} + 2 \cdot A_{toe,r} = 4800.00 + 4224.00 + 69.53 + 2 \cdot -17.38

Mass per metre length

m_{m}

71.11

\frac{kg}{m}

m_{m} = \frac{A_{t} \cdot ρ_{t}}{10^{6}} = \frac{9058.77 \cdot 7850.00}{10^{6}}

y_{web}

112.00

mm

y_{web} = t_{t} + \frac{(h_{t} - t_{t})}{2} = 24.00 + \frac{(200.00 - 24.00)}{2}

y_{flange}

12.00

mm

y_{flange} = \frac{t_{t}}{2} = \frac{24.00}{2}

y_{root}

28.02

mm

y_{root} = t_{t} + r_{r o o t} - \frac{2 \cdot r_{r o o t}}{(3 \cdot (4 - π))} = 24.00 + 18.00 - \frac{2 \cdot 18.00}{(3 \cdot (4 - π))}

y_{toe,w}

197.99

mm

y_{toe,w} = h_{t} - r_{t o e} + \frac{2 \cdot r_{t o e}}{(3 \cdot (4 - π))} = 200.00 - 9.00 + \frac{2 \cdot 9.00}{(3 \cdot (4 - π))}

y_{toe,f}

21.99

mm

y_{toe,f} = t_{t} - r_{t o e} + \frac{2 \cdot r_{t o e}}{(3 \cdot (4 - π))} = 24.00 - 9.00 + \frac{2 \cdot 9.00}{(3 \cdot (4 - π))}

Distance to centroid

c_{t}

58.38

mm

c_{t} = \frac{(A_{web} \cdot y_{web} + A_{flange} \cdot y_{flange} + A_{root,r} \cdot y_{root} + A_{toe,r} \cdot y_{toe,w} + A_{toe,r} \cdot y_{toe,f})}{A_{t}} = \frac{(4224.00 \cdot 112.00 + 4800.00 \cdot 12.00 + 69.53 \cdot 28.02 + -17.38 \cdot 197.99 + -17.38 \cdot 21.99)}{9058.77}

Second moment of area about major axis (1.1)

I_{yy,w,f}

(3.360e+7)

{mm}^{4}

I_{yy,w,f} = \frac{A_{web} \cdot {(h_{t} - t_{t})}^{2}}{12} + A_{web} \cdot {(y_{web} - c_{t})}^{2} + \frac{A_{flange} \cdot {t_{t}}^{2}}{12} + A_{flange} \cdot {(y_{flange} - c_{t})}^{2} = \frac{4224.00 \cdot {(200.00 - 24.00)}^{2}}{12} + 4224.00 \cdot {(112.00 - 58.38)}^{2} + \frac{4800.00 \cdot {24.00}^{2}}{12} + 4800.00 \cdot {(12.00 - 58.38)}^{2}

Factor for the spandrel to calculate the moment of inertia in YY

γ_{yy,local}

0.05488

γ_{yy,local} = (\frac{π}{16} - \frac{4}{(9 \cdot π)})

Second moment of area about major axis (1.2)

I_{yy,root}

69829.2

{mm}^{4}

I_{yy,root} = γ_{yy,local} \cdot {r_{r o o t}}^{4} + A_{root,r} \cdot {(y_{root} - c_{t})}^{2} = 0.05488 \cdot {18.00}^{4} + 69.53 \cdot {(28.02 - 58.38)}^{2}

I_{yy,toes}

-362559

I_{yy,toes} = - 2 \cdot γ_{yy,local} \cdot {r_{t o e}}^{4} + A_{toe,r} \cdot {(y_{toe,f} - c_{t})}^{2} + A_{toe,r} \cdot {(y_{toe,w} - c_{t})}^{2} = - 2 \cdot 0.05488 \cdot {9.00}^{4} + -17.38 \cdot {(21.99 - 58.38)}^{2} + -17.38 \cdot {(197.99 - 58.38)}^{2}

Second moment of area about major axis y

I_{yy}

(3.331e+7)

{mm}^{4}

I_{yy} = I_{yy,w,f} + I_{yy,root} + I_{yy,toes} = (3.360e+7) + 69829.2 + -362559

Second moment of area about axis x-y (part 1)

I_{xy,flange}

(-9.266e+6)

{mm}^{4}

I_{xy,flange} = A_{flange} \cdot (y_{flange} - c_{t}) \cdot (\frac{h_{t}}{2} - c_{t}) = 4800.00 \cdot (12.00 - 58.38) \cdot (\frac{200.00}{2} - 58.38)

Second moment of area about axis x-y (part 2)

I_{xy,web}

(-1.050e+7)

{mm}^{4}

I_{xy,web} = A_{web} \cdot (y_{web} - c_{t}) \cdot (\frac{t_{t}}{2} - c_{t}) = 4224.00 \cdot (112.00 - 58.38) \cdot (\frac{24.00}{2} - 58.38)

Factor for the spandrel to calculate the moment of inertia in XY

δ_{xy,local}

-0.00444

δ_{xy,local} = \frac{1}{8} - \frac{1}{(9 \cdot (4 - π))}

I_{xy,root}

63602.4

I_{xy,root} = A_{root,r} \cdot {(y_{root} - c_{t})}^{2} + δ_{xy,local} \cdot {r_{r o o t}}^{4} = 69.53 \cdot {(28.02 - 58.38)}^{2} + -0.00444 \cdot {18.00}^{4}

I_{xy,toes}

176551

{mm}^{4}

I_{xy,toes} = 2 \cdot (A_{toe,r} \cdot (y_{toe,f} - c_{t}) \cdot (y_{toe,w} - c_{t}) + δ_{xy,local} \cdot {r_{t o e}}^{4}) = 2 \cdot (-17.38 \cdot (21.99 - 58.38) \cdot (197.99 - 58.38) + -0.00444 \cdot {9.00}^{4})

Second moment of area about axis x-y

I_{xy}

(-1.953e+7)

{mm}^{4}

I_{xy} = I_{xy,flange} + I_{xy,web} + I_{xy,root} + I_{xy,toes} = (-9.266e+6) + (-1.050e+7) + 63602.4 + 176551

Second moment of area about axis u-u

I_{u u}

(5.284e+7)

{mm}^{4}

I_{u u} = I_{yy} - I_{xy} = (3.331e+7) - (-1.953e+7)

Second moment of area about axis v-v

I_{v v}

(1.378e+7)

{mm}^{4}

I_{v v} = I_{yy} + I_{xy} = (3.331e+7) + (-1.953e+7)

Radius of gyration about axis y-y

K_{yy}

60.64

mm

K_{yy} = \sqrt{\frac{I_{yy}}{A_{t}}} = \sqrt{\frac{(3.331e+7)}{9058.77}}

Radius of gyration about axis u-u

K_{u u}

76.38

mm

K_{u u} = \sqrt{\frac{I_{u u}}{A_{t}}} = \sqrt{\frac{(5.284e+7)}{9058.77}}

Radius of gyration about axis v-v

K_{v v}

39.00

mm

K_{v v} = \sqrt{\frac{I_{v v}}{A_{t}}} = \sqrt{\frac{(1.378e+7)}{9058.77}}

Elastic section modulus about the major axis

S_{yy}

235207

{mm}^{3}

S_{yy} = \frac{I_{yy}}{(h_{t} - c_{t})} = \frac{(3.331e+7)}{(200.00 - 58.38)}

Elastic section modulus about axis u-u

S_{u u}

373108

{mm}^{3}

S_{u u} = \frac{I_{u u}}{(h_{t} - c_{t})} = \frac{(5.284e+7)}{(200.00 - 58.38)}

Elastic section modulus about axis v-v

S_{v v}

97306.3

{mm}^{3}

S_{v v} = \frac{I_{v v}}{(h_{t} - c_{t})} = \frac{(1.378e+7)}{(200.00 - 58.38)}

Geometric coefficient

α_{1}

0.11272

-

α_{1} = 0.0768 + \frac{0.0479 \cdot r_{r o o t}}{t_{t}} = 0.0768 + \frac{0.0479 \cdot 18.00}{24.00}

Geometric coefficient

D_{1}

34.29

mm

D_{1} = 2 \cdot ((3 \cdot r_{r o o t} + 2 \cdot t_{t}) - \sqrt{2 \cdot {(2 \cdot r_{r o o t} + t_{t})}^{2}}) = 2 \cdot ((3 \cdot 18.00 + 2 \cdot 24.00) - \sqrt{2 \cdot {(2 \cdot 18.00 + 24.00)}^{2}})

Torsional constant

I_{T}

(1.819e+6)

{mm}^{4}

I_{T} = \frac{h_{t} \cdot {t_{t}}^{3}}{3} + \frac{(h_{t} - t_{t}) \cdot {t_{t}}^{3}}{3} + α_{1} \cdot {D_{1}}^{4} - 0.21 \cdot {t_{t}}^{4} = \frac{200.00 \cdot {24.00}^{3}}{3} + \frac{(200.00 - 24.00) \cdot {24.00}^{3}}{3} + 0.11272 \cdot {34.29}^{4} - 0.21 \cdot {24.00}^{4}

Geometric constant

g

0.85977

-

g = \sqrt{1 - \frac{I_{v v}}{I_{u u}}} = \sqrt{1 - \frac{(1.378e+7)}{(5.284e+7)}}

Equivalent slenderness coefficient

ϕ_{α}

2.50

-

ϕ_{α} = \frac{S_{u u} \cdot g}{\sqrt{A_{t} \cdot I_{T}}} = \frac{373108 \cdot 0.85977}{\sqrt{9058.77 \cdot (1.819e+6)}}